Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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Writer_Mike

29 minutes ago
In response to "Paris, I Guess"

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

QPdoll

about 17 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

2 Like

Wordsmitty ✍️

about 20 hours ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

2 Like

Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

Sep 20, 2024 at 10:13pm
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

July 24, 2023 at 9:14am

#1052996

Millimeters of Mercury

There are several articles in my queue, from various sources, about planets in our solar system. Fittingly, though randomly, the one about Mercury (from Vox) shows up here first.

Mercury was a total mystery for centuries. We're finally learning its secrets.

It's still a mystery to me. I've never knowingly seen it in the sky.

All the other visible planets, sure, no problem. Venus, when it's visible at all, is very visible, usually the brightest thing in the night sky apart from the Moon. Mars is generally easy to spot for its color; Jupiter, like Venus, tends to be bright, but isn't limited to just after sunset or just before sunrise. Saturn, while you can't see its rings with unaided eyes, is generally recognizable.

I've even seen the next planet out, through a telescope that someone else pointed. No, I won't name it until its article comes up, lest someone make a worn-out, juvenile joke about it.

But Mercury? Its visibility is even more limited than that of Venus, always close to the twilight horizon if it's visible at all, not to mention much dimmer, and that makes it tough if you don't know exactly what to look for. And despite my interest in astronomy, I never could be arsed to get one of those cellphone apps that tells you what everything is.

It's entirely possible that I've seen it and mistaken it for a star near the horizon, where some constellational context is missing.

Anyway, that's not what the article is about. Thanks to space robots (we have SPACE ROBOTS), we've seen Mercury quite a bit.

On Friday, the US Geological Survey released the most comprehensive topographical map of Mercury ever created, depicting its craters, ridges, volcanoes, and mountains — some rising more than 2 miles high — in fine detail:

You'll have to go to the link to see the map. Also, "on Friday" refers to a long-ago Friday in early 2016, as the article is pretty old now.

Because Mercury is so small and close to the sun, it's tricky to send spacecraft to visit it — before NASA's MESSENGER probe, the only craft that had come close was Mariner 10, which made a series of quick flybys in the 1970s.

The difficulty of getting to Mercury may seem counterintuitive. Woudn't a probe just fall toward the Sun? Well, no, for the same reason Earth (much to our relief) doesn't; you have to cancel a lot of angular momentum, and that takes fuel, careful planning, and lots and lots of math.

So, the rest of the article consists of some of the stuff we managed to figure out thanks to fuel, careful planning, and lots and lots of math.

1) Mercury is shrinking

And no, not because of #2 here:

2) Mercury has water ice

This has long been theorized, but lots of things were theorized that turned out not to be the case. The brief explanation for how ice can exist on the surface of a planet that's famously close to a giant perpetual fusion bomb is that Mercury has no real atmosphere to distribute heat, and the ice is in shadowed craters at the poles.

3) Mercury had a violent, volcanic past

While, again, it's good to get evidence, this isn't exactly surprising given that other terrestrial planets (and the Moon) have or had volcanic activity.

4) We can't quite figure out how Mercury formed

To be fair, we're a little fuzzy on the details of how any planets formed.

What's cool is that we're able to figure out things like how big a planet's core is and what it's made of, and that we have the technology to send robots there in the first place.

5) Mercury has a weird, off-center magnetic field

"Weird" meaning "No, we don't understand that yet, either, but it'll help us do more science."

So, there it is: stuff we didn't know before, or suspected but didn't have proof of. Always more to find out.

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